In these lessons, we will learn

• how to graph of cubic functions by plotting points.

• how to graph cubic functions of the form y = a(x − h)^{3} + k.

Related Topics: More Geometry Lessons

### How to graph cubic functions by plotting points?

The general form of a **cubic function** is y = ax^{3} + bx + cx + d where a , b, c and d are real numbers and a is not zero.

We can graph cubic functions by plotting points.

**How to graph a cubic or degree 3 polynomial function by completing a table of values?**

Complete the table using the function rule f(x) = x^{3} - 4x and graph the function.

### Graph cubic functions of the form y = a(x − h)^{3} + k.

**How to graph cubic functions by writing the function in the form y = a(x − h)**^{3} + k?

Examples: Graph

\(y = - \frac{1}{2}{\left( {x + 4} \right)^3} + 5\)

**How to graph a Transformation of a Cubic Function?**

Example:

Sketch the graph of \(f(x) = - \frac{3}{2}{\left( {x + 2} \right)^3} - 3\)**Graphing cubics using end behavior, inverted cubic, vertical shift, horizontal shift, and combined shifts**
**Graphing cubics using combined shifts, vertical stretch**

**Cubic Function Calculator**

Enter in the cubic function and select graph.

• how to graph of cubic functions by plotting points.

• how to graph cubic functions of the form y = a(x − h)

Related Topics: More Geometry Lessons

We can graph cubic functions by plotting points.

** Example: **

Draw the graph of y = x^{3} + 3 for –3 ≤ x ≤ 3. Use your graph to find

a) the value of y when x = 2.5

b) the value of x when y = –15

** Solution: **

x | –3 | –2 | –1 | 0 | 1 | 2 | 3 |

y | –24 | –5 | 2 | 3 | 4 | 11 | 30 |

a) When x = 2.5, y ≈ 18.6

b) When y = –15, x ≈–2.6

** Example: **

Plot the graph of y = x^{3} – 9x + 5 for –4 ≤ x ≤ 4 and use your graph to find:

a) the value of y when x = 1.6

b) the value of x when y = 12

** Solution: **

x | –4 | –3 | –2 | –1 | 0 | 1 | 2 | 3 | 4 |

y | –23 | 5 | 15 | 13 | 5 | –3 | –5 | 5 | 33 |

a) When x = 1.6, y ≈ –5.3

b) When y = 12, x ≈ –0.8, or x ≈ –2.5

**How to graph cubic functions using a calculator or technology?**

Complete the table using the function rule f(x) = x

We can graph cubic functions by transforming the basic cubic graph. The basic cubic graph is y = x^{3}.

For the function of the form y = a(x − h)^{3} + k.

If k > 0, the graph shifts k units up; if k < 0, the graph shifts k units down.

If h > 0, the graph shifts h units to the right; if h < 0, the graph shifts h units left.

If a < 0, the graph is flipped.

Examples: Graph

\(y = - \frac{1}{2}{\left( {x + 4} \right)^3} + 5\)

Example:

Sketch the graph of \(f(x) = - \frac{3}{2}{\left( {x + 2} \right)^3} - 3\)

Enter in the cubic function and select graph.

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